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1 6 Learning Objectives After reading this chapter, you should be able to: 1. Identify eight characteristics of the normal distribution. 2. Define the standard normal distribution and the standard normal transformation. 3. Locate proportions of area under any normal curve above the mean, below the mean, and between two scores. Probability, Normal Distributions, and z Scores 4. Locate scores in a normal distribution with a given probability. 5. Compute the normal approximation to the binomial distribution. 6. Convert raw scores to standard z scores using SPSS. Marek Uliasz / istock/ Thinkstock

2 Chapter 6: Probability, Normal Distributions, and z Scores The Normal Distribution in Behavioral Science When researchers study behavior, they find that in many physical, behavioral, and social measurement studies, the data are normally distributed. Many behaviors and individual characteristics are distributed normally, with very few people at the extremes relative to all others in a given general population. For example, most people express some level of aggression, a few are entirely passive, and a few express an abnormally high level of aggression. Most people have some moderate level of intelligence, a few score very low on intelligence, and a few score very high using the Intelligence Quotient (IQ) to measure intelligence. In Chapter 5, we introduced probability as the frequency of times an outcome is likely to occur, divided by the total number of possible outcomes. Because most behavior is approximately normally distributed, we can use the empirical rule (introduced in Chapter 4) to determine the probability of obtaining a particular outcome in a research study. We know, for example, that scores closer to the mean are more probable or likely than scores farther from the mean. In this chapter, we extend the concepts of probability to include situations in which we locate probabilities for scores in a normal distribution. In Chapter 7, we will extend the concepts of normal distributions covered in this chapter to introduce sampling distributions. 6.2 Characteristics of the Normal Distribution In 1733, Abraham de Moivre introduced the normal distribution, first discussed in Chapter 3, as a mathematical approximation to the binomial distribution, although de Moivre s 1733 work was not widely recognized until the accomplished statistician Karl Pearson rediscovered it in The shape of the curve in a normal distribution can drop suddenly at the tails, or the tails can be stretched out. Figure 6.1 shows three examples of normal distributions notice in the figure that a normal distribution can vary in appearance. So what makes a set of data normally distributed? In this section, we introduce eight characteristics that make a set of data normally distributed: Chapter Outline 6.1 The Normal Distribution in Behavioral Science 6.2 Characteristics of the Normal Distribution 6.3 Research in Focus: The Statistical Norm 6.4 The Standard Normal Distribution 6.5 The Unit Normal Table: A Brief Introduction 6.6 Locating Proportions The behavioral data that researchers measure often tend to approximate a normal distribution. The normal distribution, also called the symmetrical, Gaussian, or bell-shaped distribution, is a theoretical distribution in which scores are symmetrically distributed above and below the mean, the median, and the mode at the center of the distribution. 6.7 Locating Scores 6.8 SPSS in Focus: Converting Raw Scores to Standard z Scores 6.9 Going From Binomial to Normal 6.10 The Normal Approximation to the Binomial Distribution

3 168 Part II: Probability and the Foundations of Inferential Statistics Most behavioral data approximate a normal distribution. Rarely are behavioral data exactly normally distributed. 1. The normal distribution is mathematically defined. The shape of a normal distribution is specified by an equation relating each score (distributed along the x-axis) with each frequency (distributed along the y-axis): 1 Y = e σ 2π 2 1 x µ 2 σ It is not necessary to memorize this formula. It is important to understand that rarely do behavioral data fall exactly within the limits of this formula. When we say that data are normally distributed, we mean that the data approximate a normal distribution. The normal distribution is so exact that it is simply impractical to think that behavior can fit exactly within the limits defined by this formula. FIGURE 6.1 Frequency of Scores Three Examples of a Normal Distribution With Different Means and Standard Deviations Scores (x) σ 2 =0.2, µ=0 σ 2 =1.0, µ=0 σ 2 =0.5, µ= The normal distribution is theoretical. This characteristic follows from the first in that it emphasizes that data can be normally distributed in theory although rarely do we observe behaviors that are exactly normally distributed. Instead, behavioral data typically approximate a normal distribution. As you will see in this chapter, we can still use the normal distribution to describe behavior so long as the behaviors being described are approximately normally distributed..

4 Chapter 6: Probability, Normal Distributions, and z Scores The mean, median, and mode are all located at the 50th percentile. In a normal distribution, the mean, the median, and the mode are the same value at the center of the distribution. So half the data (50%) in a normal distribution fall above the mean, the median, and the mode, and half the data (50%) fall below these measures. 4. The normal distribution is symmetrical. The normal distribution is symmetrical in that the distribution of data above the mean is the same as the distribution of data below the mean. If you were to fold a normal curve in half, both sides of the curve would exactly overlap. 5. The mean can equal any value. The normal distribution can be defined by its mean and standard deviation. The mean of a normal distribution can equal any number from positive infinity (+ ) to negative infinity ( ): M The standard deviation can equal any positive value. The standard deviation (SD) is a measure of variability. Data can vary (SD > 0) or not vary (SD = 0). A negative standard deviation is meaningless. In the normal distribution, then, the standard deviation can be any positive value greater than The total area under the curve of a normal distribution is equal to 1.0. The area under the normal curve has the same characteristics as probability: It varies between 0 and 1 and can never be negative. In this way, the area under the normal curve can be used to determine the probabilities at different points along the distribution. In Characteristic 3, we stated that 50% of all data fall above and 50% fall below the mean. This is the same as saying that half (.50) of the area under the normal curve falls above and half of the area (.50) falls below the mean. The total area, then, is equal to 1.0. Figure 6.2 shows the proportions of area under the normal curve 3 SD above and below the mean (±3 SD). FIGURE 6.2 The Proportion of Area Within Each Standard Deviation of the Mean Area = SD 2 SD 1 SD M +1 SD +2 SD +3 SD In a normal distribution, 50% of all data fall above the mean, the median, and the mode, and 50% fall below these measures. In a normal distribution, the mean can equal any value between + and ; the standard deviation can equal any positive value greater than 0. Proportions of area under a normal curve are used to determine the probabilities for normally distributed data. The total area is equal to 1.00.

5 170 Part II: Probability and the Foundations of Inferential Statistics The tails of a normal distribution never touch the x-axis, so it is possible to observe outliers in a data set that is normally distributed The tails of a normal distribution are asymptotic. In a normal distribution, the tails are asymptotic, meaning that the tails of the distribution are always approaching the x-axis but never touch it. That is, as you travel away from the mean along the curve, the tails in a normal distribution always approach the x-axis but never touch it. Because the tails of the normal distribution go out to infinity, this characteristic allows for the possibility of outliers (or scores far from the mean) in a data set. RESEARCH IN FOCUS: THE STATISTICAL NORM Researchers often use the word normal to describe behavior in a study but with little qualification for what exactly constitutes normal behavior. Researchers studying the links between obesity and sleep duration have stated that short sleepers are at a higher risk of obesity compared to normal sleepers (see Horne, 2008; Lumeng et al., 2007), and researchers studying middle school aged children claim to measure normal changes that occur during the school year (see Evans, Langberg, Raggi, Allen, & Buvinger, 2005). What do researchers mean when they say that sleeping or change is normal? Statistically speaking, normal behavior is defined by the statistical norm, which is data that fall within about 2 SD of the mean in a normal distribution. Figure 6.3 shows that about 95% of all data fall within 2 SD of the mean in a normal distribution these data are normal only inasmuch as they fall close to the mean. Data that are more than 2 SD from the mean are characteristic of less than 5% of data in that distribution these data are not normal only inasmuch as they fall far from the mean. FIGURE 6.3 The Statistical Norm Only 5% of data fall in the tails beyond 2 SD from the mean these data are not normal or likely. 95% of data fall within 2 SD of the mean these data are normal or likely. Behavioral data that fall within 2 SD of the mean are regarded as normal (or likely) because these data fall near the mean. Behavioral data that fall outside of 2 SD from the mean are regarded as not normal (or not likely) because these data fall far from the mean in a normal distribution.

6 Chapter 6: Probability, Normal Distributions, and z Scores 171 LEARNING CHECK 1 1. All of the following characteristics are true about a normal distribution, except: a. The mean can be any positive or negative number. b. The variance can be any positive number. c. The shape of the normal distribution is symmetrical. d. The tails of a normal distribution touch the x-axis at 3 SD from the mean. 6.4 The Standard Normal Distribution In a normal distribution, the mean can be any positive or negative number, and the standard deviation can be any positive number (see Characteristics 5 and 6). For this reason, we could combine values of the mean and standard deviation to construct an infinite number of normal distributions. To find the probability of a score in each and every one of these normal curves would be quite overwhelming. As an alternative, statisticians found the area under one normal curve, called the standard, and stated a formula to convert all other normal distributions to this standard. As stated in Characteristic 7, the area under the normal curve is a probability at different points along the distribution. The standard curve is called the standard normal distribution or z distribution, which has a mean of 0 and a standard deviation of 1. Scores on the x-axis in a standard normal distribution are called z scores. The standard normal distribution is one example of a normal distribution. Figure 6.4 shows the area, or probabilities, under the standard normal curve at each z score. The numerical value of a z score specifies the distance or standard deviation of a value from the mean. (Thus, z = +1 is one standard deviation above the mean, z = 1 is one standard deviation below the mean, and so on.) Notice that the probabilities given for the standard normal distribution are the same as those shown in Figure 6.2. The probabilities are the same because the proportion of area under the normal curve is the same at each standard deviation for all normal distributions. Because we know the probabilities under a standard normal curve, we can convert all other normal distributions to this standard. By doing so, we can find the probabilities of scores in any normal distribution using probabilities listed for the standard normal distribution. To convert any normal distribution to a standard normal distribution, we compute the 2. A normal distribution has a mean equal to 5. What is the value of the median and mode in this distribution? 3. The area under a normal curve ranges between 0 and 1 and can never be negative. What type of statistic also has these same characteristics? 4. What term is often used to describe behavior that falls within 2 SD of the mean in a normal distribution? Answers: 1. d; 2. Median = 5, mode = 5; 3. Probability; 4. Normal or statistical norm. The standard normal distribution is one of the infinite normal distributions it has a mean of 0 and variance of 1. The z transformation formula converts any normal distribution to the standard normal distribution with a mean equal to 0 and variance equal to 1. The standard normal distribution, or z distribution, is a normal distribution with a mean equal to 0 and a standard deviation equal to 1. The standard normal distribution is distributed in z score units along the x-axis. A z score is a value on the x-axis of a standard normal distribution. The numerical value of a z score specifies the distance or the number of standard deviations that a value is above or below the mean.

7 172 Part II: Probability and the Foundations of Inferential Statistics Example 6.1 The standard normal transformation or z transformation is a formula used to convert any normal distribution with any mean and any variance to a standard normal distribution with a mean equal to 0 and a standard deviation equal to 1. standard normal transformation, or z transformation. The formula for the z transformation is FIGURE 6.4 z x = µ σ for a population of scores, or x M z = for a sample of scores. SD The Proportion of Total Area (total area = 1.0) Under the Standard Normal Curve M = z Scores The standard normal distribution is one example of a normal distribution. Hence, the areas in this figure are identical to those given in Figure 6.2. We use the z transformation to locate where a score in any normal distribution would be in the standard normal distribution. To illustrate, we will compute a z transformation in Example 6.1 and ask a more conceptual question in Example 6.2. A researcher measures the farthest distance (in feet) that students moved from a podium during a class presentation. The data were normally distributed with M = 12 and SD = 2. What is the z score for x = 14 feet? Because M = 12 and SD = 2, we can find the z score for x = 14 by substituting these values into the z transformation formula: z = = Figure 6.5a shows the original normal distribution of scores (x) with M = 12 and SD = 2. Notice that in Figure 6.5b, a score of x = 14 in the original distribution is exactly one z score, or one standard deviation, above the mean in a standard normal distribution.

8 Chapter 6: Probability, Normal Distributions, and z Scores 173 FIGURE 6.5 Computing the z Transformation for a Sample With M = 12 and SD = 2 6 (a) Original Distribution x = 14 is +1 SD above the mean. 16 Suppose we want to determine the z score for the mean of a normal distribution. The z transformation for the mean of any normal distribution will always equal what z score value? The mean is always at the center of a normal distribution. If you substitute the mean for x in the z transformation, the solution will always be 0. In other words, when M = x, the solution for the z transformation is The Unit Normal Table: A Brief Introduction 18 (b) Standard Normal Distribution 1 0 z = 1 SD above the mean. A score of x = 14 in the original distribution is located at z = 1.0 in a standard normal distribution, or 1 SD above the mean. The proportion of area under the standard normal distribution is given in the unit normal table, or z table, in Table B.1 in Appendix B. A portion of the table is shown in Table 6.1. The unit normal table has three columns: A, B, and C. This section will familiarize you with each column in the table. Column A lists the z scores. The table lists only positive z scores, meaning that only z scores at or above the mean are listed in the table. For negative z scores below the mean, you must know that the normal distribution is symmetrical. The areas listed in Columns B and C for each z score below the mean are the same as those for z scores listed above the mean in the unit normal table. In Column A, z scores are listed from z = 0 at the mean to z = 4.00 above the mean Example 6.2 The mean in any normal distribution corresponds to a z score equal to 0. The unit normal table or z table is a type of probability distribution table displaying a list of z scores and the corresponding probabilities (or proportions of area) associated with each z score listed.

9 174 Part II: Probability and the Foundations of Inferential Statistics TABLE 6.1 A Portion of the Unit Normal Table in B.1 in Appendix B (A) (B) (C) z Area Between Mean and z Area Beyond z in Tail Source: Based on J. E. Freund, Modern Elementary Statistics (11th edition). Pearson Prentice Hall, Column B lists the area between a z score and the mean. The first value for the area listed in Column B is.0000, which is the area between the mean (z = 0) and z = 0 (the mean). Notice that the area between the mean and a z score of 1.00 is.3413 the same value given in Figure 6.4. As a z score moves away from the mean, the proportion of area between that score and the mean increases closer to.5000, or the total area above the mean.

10 Chapter 6: Probability, Normal Distributions, and z Scores 175 Column C lists the area from a z score toward the tail. The first value for the area listed in Column C is.5000, which is the total area above the mean. As a z score increases and therefore moves closer to the tail, the area between that score and the tail decreases closer to Keep in mind that the normal distribution is used to determine the probability of a certain outcome in relation to all other outcomes. For example, to describe data that are normally distributed, we ask questions about observing scores greater than, scores less than, scores among the top or bottom %, or the likelihood of scoring between a range of values. In each case, we are interested in the probability of an outcome in relation to all other normally distributed outcomes. Finding the area, and therefore probability, of any value in a normal distribution is introduced in Section 6.6. LEARNING CHECK 2 1. A set of data is normally distributed with a mean equal to 10 and a standard deviation equal to 3. Compute a z transformation for each of the following scores in this normal distribution: (a) 2 (b) 10 (c) 3 (d) 16 (e) 0 2. Identify the column in the unit normal table for each of the following: = 3.33; 2. = 2.00, (e) z = Locating Proportions = 2.33, (d) z = The area at each z score is given as a proportion in the unit normal table. Hence, we can use the unit normal table to locate the proportion or probability of a score in a normal distribution. To locate the proportion, and therefore the probability, of scores in any normal distribution, we follow two steps: Step 1: Transform a raw score (x) into a z score. Step 2: Locate the corresponding proportion for the z score in the unit normal table. To estimate probabilities under the normal curve, we determine the probability of a certain outcome in relation to all other outcomes. (a) The z scores (b) The area from each z score toward the tail (c) The area between each z score and the mean 3. Complete the following sentence: The normal distribution is used to determine the probability of a certain outcome to all other outcomes. Answers: 1. (a) z = = 4.00, (b) z = = 0, (c) z = (a) Column A, (b) Column C, (c) Column B; 3. In relation or relative. In Examples 6.3 and 6.4, we follow these steps to locate the proportion associated with scores above the mean. In Examples 6.5 and 6.6, we follow these steps to locate the proportion associated with scores below the mean. In Example 6.7, we follow these steps to locate the proportion between two scores. In each example, we show the normal curve and shade the region under the curve that we are locating. 3 3

11 176 Part II: Probability and the Foundations of Inferential Statistics Example 6.3 For normally distributed data, we use the unit normal table to find the probability of obtaining an outcome in relation to all other outcomes. Locating Proportions Above the Mean A sample of scores is normally distributed with M = 8 and SD = 2. What is the probability of obtaining a score greater than 12? Figure 6.6 shows the normal curve for this distribution. The shaded region in Figure 6.6 represents the proportion, or probability, of obtaining a score greater than 12. We apply the two steps to find the proportion associated with this shaded region. Step 1: To transform a raw score (x) to a z score, we compute a z transformation. In this example, x = 12. The z transformation is z = = = A score equal to 12 in the distribution illustrated in Figure 6.6 is located 2.00 z scores (or 2 SD) above the mean in a standard normal distribution. FIGURE 6.6 A Normal Distribution With M = 8 and SD = M = SD above the mean Scores (x) The shaded region shows the proportion of scores that are at or above 12 in this distribution. Step 2: In this example, we are looking for the proportion from z = toward the tail. To locate the proportion, look in Column A in Table B.1 in Appendix B for a z score equal to The proportion toward the tail is listed in Column C. The proportion greater than 12 in the original distribution is p = The probability is p =.0228 of obtaining a score greater than 12.

12 Chapter 6: Probability, Normal Distributions, and z Scores 177 Example 6.4 To investigate how mindful employees are of job-related tasks, a researcher develops a survey to determine the amount of time employees at a local business spend off-task during the day; such a topic is of interest to those who study workplace behavior and performance (Dane, 2011; Wilson, Bennett, Gibson, & Alliger, 2012). After observing all employees, she reports that employees spent 5.2 ± 1.6 (M ± SD) minutes off-task during the day. Assuming the data are normally distributed, what is the probability that employees in this study spent less than 6 minutes off-task during the day? Figure 6.7 shows the distribution of times. The shaded regions in Figure 6.7 represent the proportion, or probability, of obtaining a time less than 6 minutes. We apply the two steps to find the proportion associated with the shaded region. Step 1: To transform a raw score (x) to a z score, we compute a z transformation. In this example, x = 6. The z transformation is z = = = In the distribution shown in Figure 6.7, a time equal to 6 minutes is located 0.50 z scores, or half a standard deviation, above the mean in a standard normal distribution. FIGURE 6.7 A Normal Distribution With M = 5.2 and SD = p =.5000 p = Time (in minutes) Can Stock Photo / Mark2121 The shaded region is the proportion at or below a score of 6 in this distribution.

13 178 Part II: Probability and the Foundations of Inferential Statistics The total area is.5000 above the mean and.5000 below the mean in a normal distribution. Example 6.5 Step 2: In this example, we are looking for the proportion from z = 0.50 back to the mean, and then we will add.5000, which is the total proportion of area below the mean. To locate the proportion, we look in Column A in Table B.1 in Appendix B for a z score equal to The proportion from z = 0.50 to the mean given in Column B is p = Add.1915 to the proportion below the mean (p =.5000) to find the total proportion: p = = The probability is p =.6915 that employees spent less than 6 minutes off-task during the day. Locating Proportions Below the Mean Researchers are often interested in depression across many different groups, which can be measured using the Beck Depression Inventory (BDI-II; Beck, Steer, & Brown, 1996). One such group in which depression is of interest to researchers is among groups who experience chronic pain (Lopez, Pierce, Gardner, & Hanson, 2013; Poole, White, Blake, Murphy, & Bramwell, 2009). Using data based on reports in published research, suppose a group of veterans who experience chronic pain scored 24.0 ± 12.0 (M ± SD) on the BDI-II, where higher scores indicate more severe depression. According to conventions, a score of 13 or less is regarded as a score in the minimal depression range. Assuming these data are normally distributed, what is the probability that veterans who experience chronic pain scored in the minimal depression range (13 or less)? Figure 6.8 is a normal curve showing this distribution of scores. The shaded region in Figure 6.8 is the proportion, or probability, of scores at or less than 13 on this inventory. We can follow the two steps to find the proportion associated with the shaded region. FIGURE 6.8 A Normal Distribution With M = 24.0 and SD = 12.0 p = below the mean BDI-II Scores The shaded region is the proportion at or below a score of 13 in this distribution.

14 Chapter 6: Probability, Normal Distributions, and z Scores 179 Step 1: To transform a raw score (x) to a z score, we compute a z transformation. In this example, x = 13. The z transformation is z = = = In the distribution shown in Figure 6.8, a score equal to 13 is located 0.92 z scores, or standard deviations, below the mean in a standard normal distribution. The negative sign indicates that the z score is located below the mean. Step 2: In this example, we are looking for the proportion toward the lower tail. To locate the proportion, we look in Column A in Table B.1 in Appendix B. We find z = 0.92 in the table. Again, the normal distribution is symmetrical. A proportion given for a positive z score will be the same for a corresponding negative z score. The proportion for a z score of 0.92 toward the lower tail is listed in Column C. The proportion is p = Hence, the probability is p =.1788 that a veteran given this depression measure scored in the minimal depression range (a score of 13 or less) on the BDI-II. Memory is a factor often studied in patients with trauma, such as a stroke, in which it is possible that memory is impaired (Brown, Mapleston, & Nairn, 2012; Edmans, Webster, & Lincoln, 2000). One type of metric used to measure memory among stroke patients is the Cognistat (Kiernan, Mueller, & Langston, 1987), which includes a series of tests with a higher raw score indicating more severe impairment. Using data based on reports in published research, suppose a large sample of stroke patients scored 7.5 ± 2.5 (M ± SD) on the Cognistat metric. Assuming these data are normally distributed, what is the probability that a stroke patient in this sample scored 4.1 or higher on this metric? The shaded region in Figure 6.9 reflects the proportion, or probability, of a score 4.1 or higher on this metric. We will follow the two steps to find the proportion associated with the shaded region. FIGURE 6.9 A Normal Distribution With M = 7.5 and SD = SD below the mean raw scores In the standard normal distribution, z scores above the mean are positive; z scores below the mean are negative. Example 6.6 The shaded region is the proportion at or above a score of 4.1 in this distribution.

15 180 Part II: Probability and the Foundations of Inferential Statistics Jupiter Images / Stockbyte / Thinkstock Because the normal distribution is symmetrical, probabilities associated with positive z scores are the same for corresponding negative z scores. Example 6.7 Step 1: To transform a raw score (x) to a z score, we compute a z transformation. In this example, x = 4.1. The z transformation is z = = = In the distribution shown in Figure 6.9, a score equal to 4.1 is located 1.36 z scores, or standard deviations, below the mean in a standard normal distribution. Step 2: In this example, we are looking for the proportion from z = 1.36 to the mean, and then we will add.5000, which is the total proportion of area above the mean. To locate the proportion, search Column A in Table B.1 in Appendix B for a z score equal to (Remember that a proportion given for a positive z score is the same for a corresponding negative z score.) The proportion given in Column B is p = Add.4131 to the proportion of area above the mean (p =.5000) to find the proportion: p = = The probability is p =.9131 that a patient in this sample scored 4.1 or higher on the Cognistat metric. Locating Proportions Between Two Values In recent studies, researchers have tested the possible benefits of gum chewing on academic performance in an educational environment (Hirano et al., 2008; Johnston, Tyler, Stansberry, Moreno, & Foreyt, 2012). Using data based on reports in published research, adult students who chew gum while in class improve 20 ± 9 (M ± SD) points on a standardized math test the second time they take it. Assuming these data are normally distributed, what is the probability that a student who chews gum will score between 11 and 29 points higher the second time he or she takes the standardized math test? Figure 6.10 is a normal curve showing this distribution. The shaded region in Figure 6.10 is the proportion, or probability, associated with scores between 11 and 29 on the math test. To find the proportion in the shaded regions, we apply the two steps for each score, x: Step 1 for x = 11: To transform a raw score (x) to a z score, we compute a z transformation. In this example, x = 11. The z transformation is z = = = In the distribution shown in Figure 6.10, a score equal to 11 is located 1.00 z score, or one standard deviation, below the mean in a standard normal distribution. Step 2 for x = 11: Find the z score 1.00 in Column A of Table B.1 in Appendix B, and then look in Column B for the proportion between 1.00 and the mean: p = To find the total proportion between the two scores, we will add.3413 to the proportion associated with the second score (x = 29).

16 Chapter 6: Probability, Normal Distributions, and z Scores 181 FIGURE 6.10 A Normal Distribution With M = 20 and SD = SD +1 SD below the mean above the mean Math Improvement Scores Step 1 for x = 29: Compute the z transformation for x = 29: The shaded region is the proportion of scores between x = 11 and x = 29 in this distribution z = = = A score equal to 29 in the distribution illustrated in Figure 6.10 is located 1.00 z score above the mean in a standard normal distribution. Step 2 for x = 29: The proportion between the mean and a z score of 1.00 is the same as that for 1.00: p = The total proportion between 11 and 29 is the sum of the proportion for each score: p = = The probability is p =.6826 that a student will score between 11 and 29 points higher the second time he or she takes the standardized math test. LEARNING CHECK 3 1. State the two steps for locating the proportion of scores in any normal distribution. 2. Find the probability of a score at or above the following z scores: (a) 1.23 (b) 2.50 (c) Find the probability of a score at or below the following z scores: (a) 0.08 (b) 1.00 (c) Find the probability of a score between the following z scores: (a) The mean and 1.40 (b) 1.00 and 1.00 (c).60 and 1.20 Answers: 1. Step 1: Transform a raw score (x) into a z score, Step 2: Locate the corresponding probability for that z score in the unit normal table; 2. (a) p =.1093, (b) p =.9938, (c) p =.3085; 3. (a) p =.5319, (b) p =.1587, (c) p =.9981; 4. (a) p =.4192, (b) p =.6826, (c) p =.1592.

17 182 Part II: Probability and the Foundations of Inferential Statistics Can Stock Photo Inc. / jirsak The unit normal table can be used to locate scores that fall within a given proportion or percentile. Example Locating Scores In a normal distribution, we can also find the scores that fall within a given proportion, or percentile, using the unit normal table. Finding scores in a given percentile can be useful in certain situations, such as when instructors grade on a curve with, say, the top 10% earning A s. In this example, the unit normal table can be used to determine which scores will receive an A that is, which scores fall in the top 10%. To find the cutoff score for a given proportion, we follow two steps: Step 1: Locate a z score associated with a given proportion in the unit normal table. Step 2: Transform the z score into a raw score (x). In Examples 6.8 and 6.9, we will apply these steps to locate scores that fall within a given proportion in a normal distribution. In each example, we will show the normal distribution and shade the proportion under the curve that we are given. Many researchers are interested in studying intelligence in many groups and populations, with a popular measure of intelligence being the IQ test (Castles, 2012; Spinks et al., 2007). In the general healthy population, scores on an IQ test are normally distributed with 100 ± 15 (m ± s). Based on this distribution of IQ scores, what is the minimum score required on this test to have an intelligence score in the top 10% of scores in this distribution? Figure 6.11 shows this distribution of scores. The shaded region in Figure 6.11 is the top 10% (p =.1000) of scores we need to find the cutoff or lowest score, x, in this shaded region. We will apply the two steps to locate the cutoff score for the top 10% of data: Step 1: The top 10% of scores is the same as p =.1000 toward the tail. To locate the z score associated with this proportion, we look for p =.1000 in Column C of the unit normal table in Table B.1 in Appendix B. The z score is z = A z score equal to 1.28 is the cutoff for the top 10% of data. Step 2: We need to determine which score, x, in the distribution shown in Figure 6.11 corresponds to a z score equal to Because z = 1.28, we can substitute this value into the z transformation formula: = x. 15 First, multiply both sides of the equation by 15 to eliminate the fraction: x 100 ( 15) 128. = ( 15) = x 100.

18 Chapter 6: Probability, Normal Distributions, and z Scores 183 FIGURE 6.11 Locating Scores for Example 6.8 IQ Scores Distribution x =? x = Top Look up in z table. 2 Use z transformation to solve for x. To find the solution for x, add 100 to each side of the equation: = x. A score of on the IQ test is the cutoff for the top 10% of scores in this distribution. Let us use the same IQ data for intelligence from Example 6.8, with a distribution of 100 ± 15 (M ± SD). Based on this distribution of IQ scores, what is the cutoff score for the bottom 25% of scores in this distribution? Figure 6.12 shows this distribution. The shaded region in Figure 6.12 is the bottom 25% (p =.2500). We need to find the cutoff score, x, for this shaded region. We follow the two steps to locate the cutoff score that falls in the bottom 25%. Step 1: The bottom 25% of scores is p =.2500 toward the tail. To locate the z score associated with this proportion, we look for p =.2500 in Column C of the unit normal table in Table B.1 in Appendix B. Because p =.2500 falls between z scores of 0.67 and 0.68 in the table, we compute the average of the two z scores: z = Keep in mind that this z score is actually negative because it is located below the mean. A z score equal to is the cutoff for the bottom 25% of data in this distribution. Step 2: We need to determine which score, x, in the distribution shown in Figure 6.12 corresponds to a z score equal to Because z = 0.675, we substitute this value into the z transformation formula: x = 15 Standard Normal Distribution z = 1.28 Top.1000 An IQ score greater than represents the top 10% of intelligence scores in this distribution. Steps 1 and 2 show the method used to locate the cutoff score for the top 10% of scores in this normal distribution. Example 6.9 The unit normal table can be used to locate a cutoff score for a given proportion for data that are normally distributed.

19 184 Part II: Probability and the Foundations of Inferential Statistics FIGURE 6.12 Locating Scores for Example 6.9 The unit normal table allows us to locate raw scores, x, and determine probabilities, p, for data that are normally distributed. LEARNING CHECK 4 Bottom 25% 1. State the two steps for locating the cutoff score for a given proportion of data. 2. What are the z scores associated with the following probabilities toward the tail in a normal distribution? (a).4013 (b).3050 (c).0250 (d).0505 IQ Scores Distribution First, multiply both sides of the equation by 15 to eliminate the fraction: ( 15)( ) = 100 x ( 15) = x 100. To find the solution for x, add 100 to each side of the equation. The solution given here is rounded to the nearest hundredths place: = x. Bottom 25% x =? 1 Look up in the z table. x = Use z transformation to solve for x. Standard Normal Distribution z = An IQ score less than represents the bottom 25% of intelligence scores in this distribution. Steps 1 and 2 show the method used to locate the cutoff score for the bottom 25% of scores in this normal distribution. A score equal to on the IQ test is the cutoff for the bottom 25% of scores in this distribution. 3. State the z score that most closely approximates the following probabilities: (a) Top 10% of scores (b) Bottom 10% of scores (c) Top 50% of scores Answers: 1. Step 1: Locate the z score associated with a given proportion in the unit normal table, Step 2: Transform the z score into a raw score (x); 2. (a) z = 0.25, (b) z = 0.51, (c) z = 1.96, (d) z = 1.64; 3. (a) z 1.28, (b) z 1.28, (c) z = 0.

20 Chapter 6: Probability, Normal Distributions, and z Scores SPSS in Focus: Converting Raw Scores to Standard z Scores SPSS can be used to compute a z transformation for a given data set. To demonstrate how SPSS computes z scores, suppose you evaluate the effectiveness of an SAT remedial tutoring course with 16 students who plan to retake the standardized exam. Table 6.2 lists the number of points that each student gained on the exam after taking the remedial course. We will use SPSS to convert these data into z scores. TABLE 6.2 The Number of Points Gained on the SAT Exam After Taking a Remedial Course (for 16 students) Click on the Variable View tab and enter SAT in the Name column. We will enter whole numbers, so reduce the value to 0 in the Decimals column. 2. In the Data View tab, enter the 16 values in the column labeled SAT. Go to the menu bar and click Analyze, then Descriptive Statistics and Descriptives, to display a dialog box. 3. In the dialog box, select SAT and click the arrow to move it into the Variable(s): box. Select the Save standardized values as variables box, shown in the left side of Figure Select OK, or select Paste and click the Run command. SPSS will create two outputs. First, it will create the output table shown in Table 6.3. The output table contains the sample size, the minimum and maximum score, the mean, and the standard deviation. Second, SPSS creates an additional column of z scores in the Data View tab, shown in the right TABLE 6.3 SPSS Output Table

21 186 Part II: Probability and the Foundations of Inferential Statistics FIGURE 6.13 side of Figure The added column is labeled with Z and then your variable name. In this example, we named the variable SAT, so the column is labeled ZSAT. Each z score in the ZSAT column is the number of standard deviations that the corresponding score in the original SAT column is from the mean. The Dialog Box for Step 3 (left side) and the Data View With the ZSAT Column Listing z Scores for Each of the 16 Scores (right side) MAKING SENSE STANDARD DEVIATION AND THE NORMAL DISTRIBUTION Keep in mind that the standard deviation is very informative, particularly for normally distributed data. To illustrate, consider that when students get an exam grade back, they To find the probabilities of scores often compare their in a normal distribution, you must grade to the grades know the mean and standard of others in the class. deviation in that distribution. Suppose, for example, that two professors give an exam, where the top 10% of scores receive an A grade. You take the exam given by Professor 1 and receive an 80 on the exam. Figure 6.14a shows that grades for that exam were 76 ± 2.0 (M ± SD). You then ask your friend how he did on Professor 2 s exam and find that he scored an 86 on the same exam. Figure 6.14b shows that grades for that exam were 76 ± 8.0 (M ± SD).

22 Chapter 6: Probability, Normal Distributions, and z Scores 187 FIGURE 6.14 Exam Scores in the Top 10% of a Distribution (a) Professor #1 Exam Because the mean grade is the same in both distributions, you might conclude that your friend performed better on the exam than you did, but you would be wrong. To see why, we can follow the steps to locate the cutoff score for earning an A on the exam, which is the top 10% of scores in each distribution. The lowest score you can get and still receive an A on Professor 1 s exam is the following: Step 1: The top 10% of scores is p =.1000 toward the tail located in Column C in the unit normal table. The z score associated with p =.1000 is z = Step 2: Substitute 1.28 for z in the z transformation formula and solve for x: = x. 2 Exam Scores x 76 = 2.56 (multiply both sides by 2). x = (solve for x). Your score Your score on Professor 1 s exam was an 80. Your score is in the top 10%; you earned an A on the exam. (b) Professor #2 Exam Your friend s score Exam Scores A distribution of exam scores with your grade on Professor 1 s exam (a) and your friend s grade on Professor 2 s exam (b). M = 76 in both distributions, but the standard deviations are different. The lowest score you can get and still receive an A on Professor 2 s exam is the following: Step 1: We already located this z score. The z score associated with the top 10% is z = Step 2: Substitute 1.28 for z in the z transformation equation and solve for x: = x. 8 x 76 = (multiply both sides by 8). x = (solve for x). Your friend s score on Professor 2 s exam was an 86. This score is just outside the top 10%; your friend did not earn an A on the exam. Therefore, your 80 is an A, and your friend s 86 is not, even though the mean was the same in both classes. The mean tells you only the average outcome the standard deviation tells you the distribution of all other outcomes. Your score was lower than your friend s score, but you outperformed a larger percentage of your classmates than your friend did. The standard deviation is important because it gives you information about how your score compares relative to all other scores.

23 188 Part II: Probability and the Foundations of Inferential Statistics A binomial probability distribution is distributed with µ = np and σ = npq. A binomial distribution, or binomial probability distribution, is the distribution of probabilities for each outcome of a variable with only two possible outcomes. 6.9 Going From Binomial to Normal The normal distribution was derived from the binomial distribution, which was introduced in Chapter 5. A binomial variable is one that takes on only two values. Some binomial values occur naturally; for instance, sex is male and female, and a fair coin is heads and tails. Other binomial variables are the result of categorization; for instance, height can be categorized as tall and short, and self-esteem as high and low. The distribution of random outcomes for binomial variables is called a binomial distribution, which was also introduced in Chapter 5. Binominal probabilities are distributed with m = np and σ= npq, where n = sample size, and p and q are the probabilities of each binomial outcome. From the work of Abraham de Moivre in the 1700s, we know that the outcomes for a binomial distribution approximate a normal distribution. To illustrate, suppose we select two participants from a population with an equal number of men and women. Let us construct a bar chart for the binomial distribution for selecting men to see the approximate shape of the distribution. In this example, the random variable, x, is the number of men. In a sample of two participants, there are three possible outcomes for x: 0, 1, and 2 men. Table 6.4 displays the sample space (a) and the corresponding relative frequency distribution (b) for selecting men in this population. The binomial data given in Table 6.4 are converted to a bar chart in Figure Notice that this binomial distribution appears approximately normal in shape. (a) TABLE 6.4 x The Sample Space (a) and Relative Frequency Distribution (b) for Selecting Men From a Population With an Equal Number of Men and Women Description 0 Woman Woman 1 Man Woman Woman Man 2 Man Man (b) x Relative Frequency If we increase the sample size, say, to four participants, the binomial probability distribution more clearly approximates a normal distribution. The random variable, x, in this example is still the number of men. In a sample of four participants, one of five possible outcomes could be selected for x: 0, 1, 2, 3, or 4 men. Table 6.5 displays the relative frequency distribution for selecting men in this example. The binomial data given in Table 6.5 are converted to a bar chart in Figure Notice that this binomial distribution again appears approximately normal in shape.

24 Chapter 6: Probability, Normal Distributions, and z Scores 189 FIGURE 6.15 A Bar Chart for the Relative Frequency of Selecting Men When n = 2 Relative Frequency Number of Men Selected TABLE 6.5 The Relative Frequency Distribution for Selecting Men When n = 4 x For a binomial distribution, so long as n 2, the binomial distribution will approximate a normal distribution. The larger the sample size (n), the more closely a binomial distribution will approximate a normal distribution. When n =, a binomial distribution is, in theory, a perfect normal distribution. All binomial data sets that researchers work with are less than infinite ( ) of course. So when we work with binomial data, we can use the normal distribution to approximate the probabilities of certain outcomes. Using the same notation introduced in Chapter 5, when both np and nq are greater than 10, the normal distribution can be used to estimate probabilities for binomial data. To illustrate, we use this criterion in Example Relative Frequency The binomial distribution approximates a normal distribution. The larger n is, the more closely it approximates a normal distribution.

25 190 Part II: Probability and the Foundations of Inferential Statistics FIGURE 6.16 A Bar Chart for the Relative Frequency of Selecting Men When n = 4 Example 6.10 Relative Frequency Harrell and Karim (2008) conducted a study testing sex differences in alcohol consumption in a sample of 266 women and 140 men. Can these researchers use the normal distribution to estimate binomial probabilities for sex in this sample? If both np and nq are greater than 10, then they can use the normal distribution to estimate binomial probabilities for sex. If we assign the probability of selecting a woman as p and the probability of selecting a man as q, then the probability of selecting a woman is The probability of selecting a man is Number of Men Selected 266 p = = p = = The sample size, n, is 406 adults. We can compute np and nq by substituting n, p, and q into each calculation: 2 np = 406(.66) = ; 3 4 nq = 406(.34) = Because the solution to both formulas is greater than 10, the normal distribution can be used to estimate binomial probabilities in this sample.

26 Chapter 6: Probability, Normal Distributions, and z Scores 191 LEARNING CHECK 5 1. A probability distribution for binomial outcomes approximates the shape of what type of distribution? 2. True or false: The larger the sample size, the more closely a binomial distribution approximates a normal distribution The Normal Approximation to the Binomial Distribution Assuming that np and nq are greater than 10, we can approximate binomial probabilities using the standard normal distribution. To use the normal approximation to estimate probabilities in a binomial distribution, we apply five steps: Step 1: Check for normality. Step 2: Compute the mean and standard deviation. Step 3: Find the real limits. Step 4: Locate the z score for each real limit. Step 5: Find the proportion located within the real limits. We work through these five steps in Example 6.11 to see how probabilities in a normal distribution can be used to estimate the probability of a binomial outcome. Suppose we want to estimate the probability of selecting 30 men in a sample of 50 participants from a population with an equal number of men and women. Follow the five steps to use the normal approximation to estimate probabilities for this binomial distribution. We can verify the answer we get using the five steps by comparing it to the exact binomial probability of selecting 30 men in a sample of 50 participants. The exact probability is p = Step 1: Check for normality. We first want to determine whether we can use the normal distribution to estimate probabilities for sex. In this example, n = 50, p =.50, and q =.50, where n = the total number of participants, p = the probability of selecting a man, and q = the probability of selecting a woman. We can use the normal approximation because the calculations are greater than 10: np = 50(.50) = 25; nq = 50(.50) = To use the normal approximation for the binomial distribution, np and nq must be equal to at least what value? Answers: 1. A normal distribution; 2. True; 3. Both must equal at least 10. For a binomial distribution where np and nq are greater than 10, the normal distribution can be used to approximate the probability of any given outcome. Example 6.11